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      Coherence resonance in a network of FitzHugh-Nagumo systems: Interplay of noise, time-delay, and topology

      1 , 2 , 3 , 3
      Chaos: An Interdisciplinary Journal of Nonlinear Science
      AIP Publishing

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          Impulses and Physiological States in Theoretical Models of Nerve Membrane

          Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.
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            Coherence Resonance in a Noise-Driven Excitable System

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              NEURAL EXCITABILITY, SPIKING AND BURSTING

              Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.
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                Author and article information

                Journal
                Chaos: An Interdisciplinary Journal of Nonlinear Science
                Chaos
                AIP Publishing
                1054-1500
                1089-7682
                October 2017
                October 2017
                : 27
                : 10
                : 101102
                Affiliations
                [1 ]Universitat Politècnica de Catalunya, Colom 11, Terrassa, ES-08222 Barcelona, Spain
                [2 ]Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755, USA
                [3 ]Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
                Article
                10.1063/1.5003237
                29092412
                22bab719-665b-48d0-8f09-bbfc1787a1f1
                © 2017
                History

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